TITLE: Are there 'cohomology' functors that respect all Eilenberg-Steenrod axioms except homotopy invariance?
QUESTION [9 upvotes]: What goes wrong in the axiomatic definition of a generalized (co)homology theory if one drops the axiom of homotopy invariance i.e. that homotopic maps should induce the same map in (co)homology?
Or do we have examples? Are there "interesting" or "useful" functors $\mathfrak{h}^{\cdot}:\mathrm{Spaces}\to \mathrm{Ab}$ that respect all Eilenberg-Steenrod axioms except homotopy invariance? Could such an $\mathfrak{h}$ be used to distinguish between two homotopy equivalent non-homeomorphic spaces?
(Take your favourite definition of admissible spaces)

REPLY [12 votes]: ordinary differential cohomology roughly speaking satisfies the demands of the question
the homotopy axiom fails and is replaced by a variation statement
it is a functor defined for smooth manifolds and smooth maps
the suspension isomorphism is  only true in a weaker version
and there are  [single space] axioms
Simons& Sullivan first issue of topology
finally it is quite geometric and useful in differential geometry & quantum field theory
it extends to  the generalized cohomology context 
essentially by dropping the homotopy axioms
thus it pervades a natural class of examples to the spirit of the question
this question is not obviously a good one, but it is and I salute the questioner.